Adjust 2 oven racks to the top and bottom third of an oven and preheat oven to 425 degrees F. Put the wire racks in the baking sheets and coat each generously with cooking spray.
Combine the flour, 1/2 teaspoon salt and cayenne in a large re-sealable plastic bag. Combine the eggs and 1/2 teaspoon salt in another large plastic bag; combine the breadcrumbs and 1/2 teaspoon salt in a third large plastic bag.
Add about half of the zucchini sticks to the bag with the flour and use a combination of your hands and shaking the bag to coat the zucchini. Remove the zucchini and shake off any excess flour. Transfer to the egg wash and repeat until coated. (It's less messy if you use only one hand for the dry ingredients and the other for the wet.) Let any excess egg drip off before transferring to the breadcrumb and repeat until fully coated (you'll still see some green and white of the zucchini peeking through the breading). Transfer to the prepared baking sheets and coat the breaded zucchini generously with cooking spray (there's no need to turn them). Repeat with the remaining zucchini.
Bake, rotating the baking sheets from top to bottom about halfway through, until golden brown and tender, 25 to 30 minutes.
Meanwhile, make the Buffalo sauce: Combine the hot sauce, butter and vinegar in a small saucepan and heat over medium, stirring, until all of the butter has melted and the sauce is warm. Set aside and keep warm.
Arrange the fries on a platter and drizzle them with the Buffalo sauce. Sprinkle the blue cheese and celery leaves over the top. Serve with celery and carrot sticks if desired.
Some panko breadcrumbs are coarser then others. Rub the breadcrumbs with your fingers to make them finer, which will help them coat the zucchini pieces better.
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